The Distribution of the Sample Variance Under Normality

The article translated here is actually just one part, with its own title, of a three-part paper that slays several dragons (Helmert, 1876b). Let X₁,..., X_n be independent N(µ, σ2) variates. In the omitted portions Helmert derives essentially the variance of the mean deviation % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaamOBaaaadaaeabqaamXvP5wqonvsaeHbcLgDZrNCLfwB % GyvzYfgitvMCPzgaiuaacqWF8baFaSqabeqaniabggHiLdGccaWGyb % WaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IabmiwayaaraGae8hFaWha % aa!4AAD!<![CDATA[$$\[\frac{1}{n}\sum | {X_i} - \bar X| $$ and of the mean difference % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaamOBaiaacIcacaWGUbGaeyOeI0IaaGymaiaacMcaaaWa % aabqaeaadaWgaaWcbaGaamyAaiabgcMi5kaadQgaaeqaaaqabeqani % abggHiLdWexLMBb50ujbqegiuA0nhDYvwyTbIvLjxyGmvzYLMzaGqb % aOGae8hFaWNaamiwamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadI % fadaWgaaWcbaGaamOAaaqabaGccqWF8baFaaa!5373!<![CDATA[$$\[\frac{1}{{n(n - 1)}}\sum {_{i \ne j}} |{X_i} - {X_j}|\]$$.